1. Being poor, he was always happy.

2. Being ill, he played football.
3. Being rich, he was not happy.

4. Though he was poor, he was always happy.
5. Though he was ill, he played football.
6. Though he was rich, he was not happy.

1=>4, 2=>5, 3=>6

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Are the expressions ungrammatical? In a writing test, if a student writes as in #1, #2, and #3, is it okay? In an essay, if a person writes as in #1,2 and 3, is it okay and grammatical? Or ungrammatical?

The expressions in #1, #2, and #3 are grammatically correct. They use a construction called "absolute clauses" to convey additional information about the subject of the main clause.

In this case, the absolute clauses express a condition or state (being poor, being ill, being rich) and are followed by a comma. They provide context or explanation about the subject of the main clause.

For example:
1. Being poor, he was always happy. (This means that despite his poverty, he remained happy.)
2. Being ill, he played football. (This means that even though he was sick, he still played football.)
3. Being rich, he was not happy. (This means that his wealth did not bring him happiness.)

Regarding your second question, if a student writes these expressions in a writing test or if a person includes them in an essay, they would be considered grammatically correct. However, it's important to ensure that they are used appropriately and effectively to convey the intended meaning.